The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 18, Number 1 (2008), 209-244.
The lineage process in Galton–Watson trees and globally centered discrete snakes
We consider branching random walks built on Galton–Watson trees with offspring distribution having a bounded support, conditioned to have n nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of “globally centered discrete snake” that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when n goes to +∞, “globally centered discrete snakes” converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton–Watson tree conditioned by the size, and their links with a multinomial process [the lineage of a node u is the vector indexed by (k, j) giving the number of ancestors of u having k children and for which u is a descendant of the jth one]. Some consequences concerning Galton–Watson trees conditioned by the size are also derived.
Ann. Appl. Probab., Volume 18, Number 1 (2008), 209-244.
First available in Project Euclid: 9 January 2008
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Marckert, Jean-François. The lineage process in Galton–Watson trees and globally centered discrete snakes. Ann. Appl. Probab. 18 (2008), no. 1, 209--244. doi:10.1214/07-AAP450. https://projecteuclid.org/euclid.aoap/1199890021